# Application of the Spherical Shaping Method to a Low-Thrust Multiple Asteroid Rendezvous Mission

### Implementation, limitations and solutions

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## Abstract

Since the development of the exponential sinusoid for low-thrust trajectory design by Petropoulos and Longuski [2004], more shape-based methods have emerged that ease up the work of a mission analyst. These methods are the ideal tool to generate a quick and almost complete overview of a large search space and are useful for producing first estimate trajectories. The spherical shaping method by Novak [2012] is one of the more recently developed methods capable of shaping a transfer in three dimensions, satisfying constraints on initial and final position and velocity, and capable of satisfying a time of flight constraint; all at the same time. This makes it a particularly interesting method for application to rendezvous missions. In this thesis, the spherical shaping method will be used to generate first estimate trajectories for the GTOC2-mission (a multiple asteroid rendezvous mission). The goal is to find out whether or not the spherical shaping method is capable of producing sub-optimal trajectories. Implementing the spherical shaping method turned out to be a more massive job than anticipated. The documentation in the PhD thesis by Novak [2012] appeared not to be fully sufficient to use the method. Additional derivations were necessary to grasp the meaning of some functions. In this thesis, this process is explained and corrections of some typos found in Novak’s PhD thesis, are given. The spherical shaping function relations are re-ordered in a step-wise implementation scheme. This helps to get an overview of the implementation and will make future applications of this method easier. The implemented method is fully validated, both with respect to general two- and three-dimensional trajectories and with respect to the same test cases as used by Novak [2012]. The applicability of any method is dependent on its limits. For the spherical shaping method, these limits were not given unambiguously. Novak [2012] speaks of low and high inclinations but does not specify what is meant by "low" and "high". To find out what the limits for the inclination of the orbit are, several trajectories were computed for a range of inclination angles, for both trajectories at a constant inclination and trajectories with inclination changes. It was found that up until an inclination of 15 degrees, the relative difference between the deltav for a transfer at this constant inclination and the deltav for a same transfer at a zero inclination remains below 1 percent. For higher inclinations the difference rises quickly. At high inclinations of about 50 degrees the method breaks down. Also the Keplerian arc can not be reconstructed at inclined orbits. A way to solve this problem is to perform a reference frame transformation. A transformation involving a rotation over the line of nodes to remove the initial inclination was developed. This transformation works perfectly when the right ascension of both the initial and final orbit is equal and removes the effects caused by the high inclination. Also for high inclinations of above 50 degrees the spherical shaping method with reference frame transformation keeps producing feasible trajectories. The Keplerian arc can now be reconstructed at any inclination. When the right ascension of both orbits differs, the rotation is performed over the initial line of nodes, which makes it a less interesting rotation for the final orbit. To find out up to which right ascension difference the transformation can be used, several trajectories were computed for a range of right ascension differences (keeping other characteristics equal). It was found that up to 10 degrees the reference frame transformation can be useful. For higher differences in the initial and final right ascension of ascending node, the reference frame transformation could do more harm than good. To solve for high inclination changes, a solution was proposed to solve for multiple smaller inclination changes and sum the results. A promising result is obtained when combining this summation with the reference frame transformation. A last verification is performed by applying the implemented spherical shaping method to multiple test cases. Good results are obtained and therefore the implementation is considered fully validated. Finally the spherical shaping method is applied to the GTOC2 problem, using the top 3 asteroid combinations found by GTOC2, Heiligers [2013] and Secretin [2012]. Monte Carlo simulations with 100,000 samples were run and for each asteroid combination feasible trajectories are obtained, within the constraints set by GTOC2. The initial search space was set too broad which resulted in less feasible trajectories for GTOC2. Additional runs for a smaller search space are needed. There is however no time left to do this as well. It is recommended that the search is continued and a more thorough optimisation is performed. For the objective function of the GTOC2 problem a best value of 71.12 kg/yr was obtained for the asteroid combination equal to GTOC2 rank 2. Also a minimum value of deltav was obtained, equal to 20.04 km/s, for the asteroid combination of Secretin [2012] rank 3. Better results for both the objective function values or deltav for all asteroid combinations are expected when a more extensive optimisation is performed for the implementation of the spherical shaping method.